Shape-based segmentation methods

The shape of an object is an abstract concept and is based on human perception which is quite variable. Shapes are generally represented using contours, transforms, or regions [20]. Contour-based methods represent the shape outline either using a set of points on the contour or approximate the curve using a function such as the level-set function. Region-based methods may partition a shape into simpler forms (such as polygons), approximate the shape using a bounding region (such as a bounding rectangle or convex hull), or represent internal features of the shape (e.g., a skeleton). Transform- based representations decompose a shape into one or two dimensional signals (for example Fourier and wavelet transforms are linear transforms, whereas the Hough transform is a nonlinear transform). Transform-domain descriptors of shape can be transform coefficients or transform energy. In this project, shape has been represented using a contour because it can be easily deformed to represent a flexible boundary.

18

When an object is enlarged, rotated or moved it is still recognizable by a human. This property of an object is called pose invariance i.e., the object is identifiable from a different angle or position, or at a different scale. The pose of an object in an image can be changed using an affine transform. Pose is a relative concept and it is usually calculated with respect to the pose of another similar object in an image. Pose can be estimated by deriving the parameters of the affine transform needed to match the two shapes.

A. Active contour method of segmentation

Deformable contour models or active-contour models are shape-based procedures where a closed contour deforms using an energy function. This energy function incorporates regional properties of the object, such as edges or mean pixel intensity, and/or object-level features such as curvature of the object and size. During the curve evolution process, the curve is driven towards the boundary of the object. One approach for curve evolution is the marker point method [83] in which a segmenting curve C is parameterized by converting each point on the curve to represent a position vector

( , ), ( , )

x s t y s t , where s are points of the curve along a certain orientation (clockwise or counterclockwise), and t is time. The simplest way to evolve the segmenting curve is to move it along its normal vector field. Thus, if C is a simple, closed and smooth initial curve in2, it moves with a speed F along its normal direction,

 

19

is the unit normal to the curve. This ordinary differential equation is called the “Lagrangian” form of the equation of motion. A standard procedure for computing the evolution of the moving front is to discretize this Lagrangian equation of motion. This is performed by dividing the curve into M equal (or possibly unequal) mesh points

, 0,...,

i

s  i s i M of size s , and time into n equal intervals of length t producing M+1 marker points on the curve (or interface). Figure 2.4 shows a discrete parameterization of a curve and normal vectors showing the direction of curve evolution. The front can be interpolated from these marker points as either line segments in two- dimensions (2D) or triangles in three-dimensions (3D). The curve evolution produces points for the (n+1)th time ( ( ,x i n1), ( ,y i n1)) from the previous positions at time n on the interface. One disadvantage of this method is that the curve evolution can lead to two marker points coming closer to each other into a corner and within a few time steps this can lead to oscillations in the curvature, making the output unbounded.

s   2 s   8 s   7 s   4 s   1 s   6 s   3 s   5

20

Another approach to active shape modeling is the level-set method introduced by Sethian [70][92]. This methodology became very popular due to its ability to automatically represent changes in the topology of dynamic curves such as the boundaries of soap bubbles, flames and other physical phenomena whose shape changes with time. In this approach the evolving boundary (interface) is represented implicitly as the zero isocontour of some function. For example, the zero isocontour of

2 2 ( )x x y 1

    is given by a unit circle defined by( )x 0. In this framework, the equation of motion of the front is defined using a simple convection equation such as:

0,

V

   (2.7)

where  is the temporal partial derivative of the implicit function, V u v w, , is the velocity field (u, v, w are components of the velocity field in the x, y and z directions respectively), and  is the spatial gradient operator such that V  u_xv_yw_z This formulation is called the “Eulerian” approach since the interface is represented by an implicit function as opposed to marker points in the Lagrangian formulation. Equation (2.7) is referred to as the level set equation.

The level set function is usually defined in terms of the signed distance function. The signed distance function is an implicit function that takes any point in the plane and returns as its output the Euclidean distance between that point and the closest point on the interface. Pixels outside the interface have positive distance while the pixels inside have negative distance values assigned to them (figure 2.5). The “zero” level set is defined as the interface itself, i.e., the set of all points that are at height zero, or equivalently, whose distance to the interface is zero.

21

The level set update equation is derived by discretizing the level set equation using the forward Euler time discretization given by:

( 1) ( ) ( ) ( ) 0. n n V n n t    _{  }  (2.8)

The spatial derivative terms in equation (1) can be expanded as:

( 1) ( ) ( ) x( ) ( ) y( ) ( ) z( ) 0. n n u n n v n n w n n t    _{  }      (2.9)

The upwind differencing used for the spatial derivative terms along with the forward Euler time discretization makes the level set update stable. This guarantees that small approximation errors are not amplified with time.

Figure 2.5. Signed distance representation of a contour with negative distances assigned to pixels inside the contour and positive distances assigned to pixels outside.

22

B. Level set methods for medical image segmentation

Level set methods have been used extensively for medical image segmentation [21]. Some of the popular methods are by Leventon et al.[64], Tsai et al. [99] and Chan and Vese [17]. Leventon et al. introduced the concept of shape representation by principal component analysis (PCA) on signed distance functions. They also incorporated statistical shape priors into their geodesic active-contour model to generate maximum a posteriori estimates of pose and shape. They segmented synthetic as well as medical images using their method and compared level-set evolution with and without shape influence. Their segmentation results were within one or two voxels of manual segmentation. However, the initialization point was placed manually on the images.

Tsai et al. derived a shape-based level set function. Tsai et al.‟s goal was to find the parameters of this function that produce a good model of the object shape based on priors from the training data. Tsai et al. derived these parameters via an optimization procedure that used statistics defined over local regions in a set of training images. The performance of Tsai et al.‟s algorithm thus depended on the particular choice of statistics used to distinguish various regions within a given image. They showed automatic segmentation results on several synthetic images and semi-automatic segmentation on cardiac and pelvic MRI images. The goal of the Tsai et al. algorithm was to optimize parameters of the level set function using the statistics of pixel intensity values in images. The LSGA also optimizes the parameters of the same level set function, but using a GA, allowing high-level texture and spatial relationships to be used for optimization.

Shape priors have also been used with active-contour-based image segmentation by Etyngier et al. [30]. They used diffusion maps to model shapes as finite-dimensional

23

manifolds. Their segmentation results were accurate but the initial contour was placed manually in the images.

Chan & Vese introduced a region-based energy function based on Mumford-Shah segmentation techniques [80] in order to detect features with diffuse boundaries. The limitation of their model is that it could only detect objects by intensity average values. They mention that other image features such as texture need to be combined with a level- set framework in the future to perform more generalized Mumford-Shah segmentation. The LSGA developed here attempts to address this need. The performance of the LSGA is compared with the Chan & Vese algorithm in a later chapter; here we describe the Chan & Vese algorithm.